Which of the two results is the correct approach to compute the matrix-product AB ???

Does there exist a rule in linear algebra which allows me to predetermain if the product of two matrices A and B both not of the same size ( A is n x n and B is m x n ) gives the resulting matrix C which has a different size than A and B ???

If a matrix A is n x n and a matrix B is m x n then the matrix-product AB does't exist at all. You can compute only BA.
In general. If A is m x k, and B is k x n, than AB is m x n. But BA isn't defined. Number of columns of first matrix must be equal to number of rows of second